December 1, 2008

In Memoriam J. R. Stallings – Topology of Finite Graphs

In light of the sad news about John Stallings reported by Jesse, I thought I’d remember him by recounting my favourite proof. I don’t know if I love it because it inspired my thesis or vice versa, but it’s been an important part of my mathematical life so far.

In ‘Topology of Finite Graphs’ [1], Stallings exploited the observation that free groups are precisely the fundamental groups of graphs to give elementary topological proofs of lots of classical theorems about subgroups of free groups. This simple idea enables ignoramuses like me, whose group theory is pretty ropey, to prove nice theorems about free groups using nothing more than undergraduate topology!

The paper is a goldmine: among various highlights, Stallings gives a proof of Howson’s Theorem that the intersection of two finitely generated subgroups of a free group is finitely generated, and I think it’s the first place that his famous folds appear. The theorem I want to focus on was first proved by Marshall Hall Jr in 1949 .

Theorem(M. Hall Jr): If F is a free group and H is a finitely generated subgroup then H is a free factor in a finite-index subgroup of F.

I’d like to outline Stallings’ proof. As we go, we’ll see that this theorem is as much a statement about the topology of graphs as it is about group theory. The only prerequisite should be covering space theory. Here goes.

The ambient free group F may as well be finitely generated – any finitely generated subgroup is contained in a finitely generated free factor. So, as in all these proofs, we identify F with the fundamental group of some finite graph. For definiteness, if F is of rank n then let G be the ‘rose’ with n petals – that is, the graph with one vertex and n edges. To keep track of things, let’s orient the edges of G – in other words, put an arrow on each edge, so one end is leaving the vertex and the other is entering it.

The idea of the proof is to analyse immersions into G. Recall that a map of graphs f from D to G (sending vertices to vertices and edges to edges) is an immersion if it is locally injective – so whenever v is a vertex of D, f embeds a neighbourhood of v into a neighbourhood of f(v). If f is a local homeomorphism then f is a covering map. The next theorem is the heart of Stallings’ proof.

Theorem(Stallings): For any immersion f from a finite graph D to G there is a finite-sheeted covering space D’ of G that extends f. More precisely, there is an embedding of D into D’ and the restriction of the covering map to D coincides with f.

The proof of this theorem is a fun exercise. The idea is to look at the vertices of D in turn and to count the number of ‘missing’ edges that need to be included to turn f into a covering. Here’s an example. The colours and arrows are meant to indicate where each map sends each edge. So red edges get sent to red edges, in such a way that the arrows match up. The dotted edges are the ones that need to be added to obtain a covering space.

Armed with Stallings’ theorem, Marshall Hall’s Theorem is quite easy.

Proof of Marshall Hall’s Theorem: As above, identify F with the fundamental group of G. Let G’ be the covering space corresponding to the subgroup H. Now G’ is an (in general infinite) graph, but because H is finitely generated, its fundamental group is carried by a finite, connected subgraph D. Pictorially, G’ looks like D with some infinite trees attached.

The restriction of the covering map to D is an immersion. By Stallings’ theorem, D can be extended to a finite-sheeted covering space D’. This corresponds to a finite-index subgroup K of F, and as H is carried by the embedded subgraph D, it corresponds to a free factor in K. QED

So go forth, topologists, and prove theorems about free groups!

Remarks: 1. These ideas were floating around for a while before the paper appeared, and I believe a similar proof is given in Hempel’s 3-manifolds book. I did some investigation while writing my thesis, and there seems to be general agreement that these ideas are essentially due to Stallings.

2. Stallings’ ideas are the beginning of a long story. The first and perhaps most notable proof in a similar vein is Scott’s theorem that finitely generated subgroups of surface groups are virtually geometric [2]. His argument makes really elegant use of hyperbolic geometry. Haglund and Wise’s ‘Special cube complexes’ [3] is a very recent highlight. It seems to simultaneously unite Scott’s and Stallings’ approaches and to open the door to proving similar theorems for much more general classes of groups. I may write something about this in the future.

3. For any topologists out there wondering why they should care about free groups, the corresponding theorem for 3-manifolds would imply that every immersed surface can be lifted to an embedding in a finite-sheeted covering space. There are examples of graph manifolds where this doesn’t hold, but the question is still open for hyperbolic 3-manifolds. The link to the Virtual Haken Conjecture is obvious.

I gave a talk about the analogous theorem for residually free groups [4] at MSRI in the autumn, and was really gratified when Stallings came to talk to me afterwards about it. He was an idol of mine, and I’m sad not to have had the opportunity to talk maths with him more often.